Exact Solutions of the Generalized Benjamin-Bona-Mahony Equation

We apply the theory of Weierstrass elliptic function to study exact solutions of the generalized Benjamin-Bona-Mahony equation. By using the theory of Weierstrass elliptic integration, we get some traveling wave solutions, which are expressed by the hyperbolic functions and trigonometric functions. This method is effective to find exact solutions of many other similar equations which have arbitrary-order nonlinearity. 1. Introduction The nonlinear phenomena in the scientific work or engineering fields are more and more attractive to scientists. To depict and analyze such nonlinear phenomena, the nonlinear evolutionary equations are playing an important role and their solitary wave solutions are the main interests of mathematicians and physicists. To obtain the traveling wave solutions of these nonlinear evolution equations, many methods were attempted, such as the inverse scattering method, Hirota’s bilinear transformation, the tanh-sech method, extended tanh method, sine-cosine method, homogeneous balance method, and exp-function method. With the aid of symbolic computation system, many explicit solutions are easily obtained, and many interesting works deeply promote the research of nonlinear phenomena. The present work is interested in generalized Benjamin-Bona-Mahony (BBM) equation: In the above equation, the first term of left side represents the evolution term while parameters and represent the coefficients of dual-power law nonlinearity, and are the coefficients of dispersion terms, is the power law parameter, and variable is the wave profile. In [1], Biswas used the solitary wave ansatz and obtained an exact 1-soliton solution of (1.1). In order to find more exact solutions of some nonlinear evolutionary equations, the Weierstrass elliptic function was introduced. For example, Kuru [2, 3] discussed the BBM-like equation, and Estévez et al. [4] analyzed another type of generalized BBM equations. In [5], Deng et al. also applied the similar method to the study of a nonlinear variant of the PHI-four equation. In this paper, we will apply the method to the generalized Benjamin-Bona-Mahony equation. The rest of this paper is organized as follows. In Section 2, we first outline the Weierstrass elliptic function method. In Section 3, we give exact expression of some traveling wave solutions of generalized Benjamin-Bona-Mahony (BBM) equation (1.1) by using the Weierstrass elliptic function method. Finally, some conclusions are given in Section 4. 2. Description of the Weierstrass Elliptic Function Method When we search for the solutions of some